lad.ipynb

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   "cell_type": "markdown",
   "id": "1ae63b19-671d-4efd-a8af-fe1a7977d1d2",
   "metadata": {},
   "source": [
    "# Estimating LAD (Least Absolute Deviations) Regression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "6dad5977-5889-4821-92b5-0961c91984d9",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Info: Loading package JuMP\n",
      "└ @ Main In[1]:1\n",
      "┌ Info: Loading package GLPK\n",
      "└ @ Main In[1]:3\n",
      "┌ Info: Loading Plots\n",
      "└ @ Main In[1]:5\n"
     ]
    }
   ],
   "source": [
    "@info \"Loading package JuMP\"\n",
    "using JuMP\n",
    "@info \"Loading package GLPK\"\n",
    "using GLPK\n",
    "@info \"Loading Plots\"\n",
    "using Plots"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9a441c0c-a8a5-4e0b-9658-1cbc03063f2f",
   "metadata": {},
   "source": [
    "## Phone Data "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "ba74f425-58ba-4f53-aedc-33c42267a629",
   "metadata": {},
   "outputs": [],
   "source": [
    "year = collect(50:73)\n",
    "calls=[4.4, 4.7, 4.7, 5.9, 6.6, 7.3, 8.1, 8.8, 10.6, 12.0, 13.5, 14.9, 16.1, 21.2, 119.0, 124.0,\n",
    "          142.0, 159.0, 182.0, 212.0, 43.0, 24.0, 27.0, 29.0];"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f05497ec-cd96-45fa-a2d8-f8651add42e9",
   "metadata": {},
   "source": [
    "## Scatter Plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "a8f093cf-7c14-4efa-af36-b9b5222d1408",
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "scatter(year, calls)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fa494184-9d23-4492-b12d-fe47696802d6",
   "metadata": {},
   "source": [
    "## Goal Programming Model"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "caface34-1087-4c26-8d43-9a553b71f36b",
   "metadata": {},
   "source": [
    "$$\n",
    "\\begin{aligned}\n",
    "\\min z & = \\sum_{i=1}^{n} d_i^{-} + d_i^{+}\\\\\n",
    "\\text{subject to:} \\\\\n",
    "& y[i] - \\beta_0 - \\beta_1 x[i] + d_1^{-} - d_1^{+} = 0 \\\\\n",
    "& d_i^{-}, d_i^{+} \\ge 0 \\\\\n",
    "& \\beta_0, \\beta_1 \\in  \\mathbb{R} \\\\\n",
    "& i = 1, 2, \\ldots, n\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "eb604fa3-7587-4c90-87f1-6c1d75cdeff6",
   "metadata": {},
   "source": [
    "## Creating Model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "2041154b-dad3-492d-880d-b3f63b9c9784",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Info: Creating model\n",
      "└ @ Main In[4]:1\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "A JuMP Model\n",
       "Feasibility problem with:\n",
       "Variables: 0\n",
       "Model mode: AUTOMATIC\n",
       "CachingOptimizer state: EMPTY_OPTIMIZER\n",
       "Solver name: GLPK"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@info \"Creating model\"\n",
    "m = Model(GLPK.Optimizer)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "eb34e78a-0df4-400a-900a-4351906a2bce",
   "metadata": {},
   "source": [
    "## Number of observations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "7ebcbf07-cf72-4c9f-974b-36751fb0e720",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Info: Length\n",
      "│   n = 24\n",
      "└ @ Main In[5]:2\n"
     ]
    }
   ],
   "source": [
    "n = length(year)\n",
    "@info \"Length\" n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "174bd9fd-8133-45e8-bee6-7f30451b39d3",
   "metadata": {},
   "source": [
    "## Creating Decision Variables"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "08497182-6a21-4b4a-955a-40c81c21fd8f",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Info: Creating variables\n",
      "└ @ Main In[6]:1\n"
     ]
    }
   ],
   "source": [
    "@info \"Creating variables\"\n",
    "@variable(m, d[1:(2n)])\n",
    "@variable(m, beta0)\n",
    "@variable(m, beta1);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b86b0830-44be-4ac0-80f0-2c5ec71c40ce",
   "metadata": {},
   "source": [
    "## Creating objective function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "c19448e8-9f8a-4147-bd3f-5754b9d73ca9",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Info: Creating Goal function\n",
      "└ @ Main In[7]:1\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$$ d_{1} + d_{2} + d_{3} + d_{4} + d_{5} + d_{6} + d_{7} + d_{8} + d_{9} + d_{10} + d_{11} + d_{12} + d_{13} + d_{14} + d_{15} + d_{16} + d_{17} + d_{18} + d_{19} + d_{20} + d_{21} + d_{22} + d_{23} + d_{24} + d_{25} + d_{26} + d_{27} + d_{28} + d_{29} + d_{30} + d_{31} + d_{32} + d_{33} + d_{34} + d_{35} + d_{36} + d_{37} + d_{38} + d_{39} + d_{40} + d_{41} + d_{42} + d_{43} + d_{44} + d_{45} + d_{46} + d_{47} + d_{48} $$"
      ],
      "text/plain": [
       "d[1] + d[2] + d[3] + d[4] + d[5] + d[6] + d[7] + d[8] + d[9] + d[10] + d[11] + d[12] + d[13] + d[14] + d[15] + d[16] + d[17] + d[18] + d[19] + d[20] + d[21] + d[22] + d[23] + d[24] + d[25] + d[26] + d[27] + d[28] + d[29] + d[30] + d[31] + d[32] + d[33] + d[34] + d[35] + d[36] + d[37] + d[38] + d[39] + d[40] + d[41] + d[42] + d[43] + d[44] + d[45] + d[46] + d[47] + d[48]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@info \"Creating Goal function\"\n",
    "@objective(m, Min, sum(d[i] for i in 1:(2n)))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cdcd87cf-2c8d-404e-83fc-6bd9b2f1d740",
   "metadata": {},
   "source": [
    "## Creating constraints"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "b6fa44a3-55c4-4af5-a663-dde6d4ffd3ac",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Info: Creating constraints\n",
      "└ @ Main In[8]:1\n"
     ]
    }
   ],
   "source": [
    "@info \"Creating constraints\"\n",
    "for i in 1:n\n",
    "    @constraint(m,  calls[i] - beta0 - beta1 * year[i] + d[i] - d[n + i] == 0)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "042120d0-c5da-4aa2-9adf-c1334b91e621",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Info: Adding sign constraints\n",
      "└ @ Main In[9]:1\n"
     ]
    }
   ],
   "source": [
    "@info \"Adding sign constraints\"\n",
    "for i in 1:(2n)\n",
    "    @constraint(m, d[i] >= 0)\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "99cc8fae-bbcc-4c95-99e8-d8465b75a13c",
   "metadata": {},
   "source": [
    "## Solve!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "570be8e4-6e9e-4907-aa27-79e7757a1cf0",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Info: Solving\n",
      "└ @ Main In[10]:1\n"
     ]
    }
   ],
   "source": [
    "@info \"Solving\"\n",
    "optimize!(m)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "789b0075-2d7a-4b78-bffc-a9ff7914b5bb",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Info: -75.18999999999991\n",
      "└ @ Main In[11]:1\n",
      "┌ Info: 1.529999999999998\n",
      "└ @ Main In[11]:2\n"
     ]
    }
   ],
   "source": [
    "@info value(beta0)\n",
    "@info value(beta1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c26cb940-1dad-41e2-928b-8624e48ed271",
   "metadata": {},
   "source": [
    "## Estimated Regression Model"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fa376681-676c-4a6f-9a71-6c5cf7860d8a",
   "metadata": {},
   "source": [
    "$$\n",
    "\\hat{y} = -75.18999 + 1.5299 x \n",
    "$$"
   ]
  },
  {
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   "execution_count": 16,
   "id": "3a2a0845-e682-4207-9f05-d977195eef3f",
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     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "scatter(year,calls)\n",
    "Plots.abline!(1.5299, -75.18999)"
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  {
   "cell_type": "code",
   "execution_count": null,
   "id": "c239796c-019e-469e-a1b6-f04ec27ce6e7",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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